A geometric approach to quantum control in projective hilbert spaces

As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.

In this work, we have studied the non-Hermitian nonlinear Landau–Zener–Stückelberg–Majorana (LZSM) interferometry in a non-Hermitian N-body interacting boson system in which the non-Hermiticity is from the nonreciprocal tunnelings between the bosons. By using the mean-field approximation and projective Hilbert space, the effect of nonreciprocity and nonlinearity on the energy spectrum, the dynamics, and the formation of the interference fringes have been studied. The different symmetries and the impact of the two different types of reciprocity, i.e. the in-phase tunneling and anti-phase tunneling, on the energy spectrum and the phase transition between the Josephson oscillations and the self-trapping have been investigated. For the LZSM interferometry, the strength of the nonreciprocity is found to take an essential role in the population of the projective state and the strengths of the interference patterns in the projective space. While the conditions of destructive and constructive interference under the weak-coupling approximation still only depend on the strength of nonlinearity. Our result provides an application of the nonlinear non-Hermitian LZSM interferometry in studying the parameters of a two-level system which related to the nonlinearity and the non-Hermiticity.

Global feedback stabilizability results are derived for nonautonomous coupled systems arising from the linearization around a given time-dependent trajectory of FitzHugh-CNagumo type systems. The feedback is explicit and is based on suitable oblique (nonorthogonal) projections in Hilbert spaces. The actuators are, typically, a finite number of indicator functions and act only in the parabolic equation. Subsequently, local feedback stabilizability to time-dependent trajectories results are derived for nonlinear coupled parabolic-ODE systems of the FitzHugh-CNagumo type. Simulations are presented showing the stabilizing performance of the feedback control.

The minimum distance approach for reconstructing a positive function based on knowledge of finitely many linear functional values is examined. Two important classes of directed distances for signal processing and statistical inference are discussed. By imposing conditions analogous to those satisfied by linear projections in Hilbert space, two logarithmic entropy principles are derived. One of these involves the Itakura–Saito distortion measure of communication theory and uniquely extends Burg's maximum entropy method to incorporate prior knowledge. The other uses the Kullback–Leibler distance of statistics.

Polynomial estimates for the method of cyclic projections in Hilbert spaces

This paper addresses a new class of differential variational inequalities that have recently been introduced and investigated in finite dimensions as a new modeling paradigm of variational analysis to treat many applied problems in engineering, operations research, and physical sciences. This new subclass of general differential inclusions unifies ordinary differential equations with possibly discontinuous right-hand sides, differential algebraic systems with constraints, dynamic complementarity systems, and evolutionary variational systems. The purpose of this paper is two-fold. Firstly, we show that these differential variational inequalities, when considering slow solutions and the more general level of a Hilbert space, contain projected dynamical systems, another recent subclass of general differential inclusions. This relation follows from a precise geometric description of the directional derivative of the metric projection in Hilbert space, which is based on the notion of the quasi relative interior. Secondly we are concerned with stability of the solution set to this class of differential variational inequalities. Here we present a novel upper set convergence result with respect to perturbations in the data, including perturbations of the associated set-valued maps and the constraint set. Here we impose weak convergence assumptions on the perturbed set-valued maps, use the monotonicity method of Browder and Minty, and employ Mosco convergence as set convergence. Also as a consequence, we obtain a stability result for linear complementarity systems.

This paper is concerned with the shape sensitivity analysis of a class of boundary control, constrained, optimal control problems for parabolic systems. The notion of Euler and Lagrange derivatives of a boundary optimal control in the direction of a vector field is introduced. The derivatives are obtained in the form of optimal solutions of auxiliary optimal control problems. The method of sensitivity analysis used in this paper is based on related results on the differential stability of metric projections in Hilbert space onto a convex, closed subset, combined with the material derivative method of shape sensitivity analysis. Parabolic initial-boundary value problems with Dirichlet and Neumann boundary conditions are considered in this paper.

A quantum system interacting with its environment is subject to dephasing, which ultimately destroys the information it holds. Here we use a superconducting qubit to experimentally show that this dephasing has both dynamic and geometric origins. It is found that geometric dephasing, which is present even in the adiabatic limit and when no geometric phase is acquired, can either reduce or restore coherence depending on the orientation of the path the qubit traces out in its projective Hilbert space. It accompanies the evolution of any system in Hilbert space subjected to noise.

Let X be a real Banach space with an unconditional basis (for example, X = ℓ2 Hilbert space), let Ω ⊂ X be open, let M ⊂ Ω be a closed split real analytic Banach submanifold of Ω, let E → M be a real analytic Banach vector bundle, and let 𝒜 E → M be the sheaf of germs of real analytic sections of E → M. We show that the sheaf cohomology groups Hq(M, 𝒜 E) vanish for all q ⩾ 1, and there is a real analytic retraction r:U → M from an open set U with M ⊂ U ⊂ Ω such that r(x) = x for all x ∈ M. Some applications are also given, for example, we show that any infinite-dimensional real analytic Hilbert submanifold of separable affine or projective Hilbert space is real analytically parallelizable.

Hughston has recently proposed a stochastic extension of the Schrödinger equation, expressed as a stochastic differential equation on projective Hilbert space. We derive new projective Hilbert space identities, which we use to give a general proof that Hughston’s equation leads to state vector collapse to energy eigenstates, with collapse probabilities given by the quantum mechanical probabilities computed from the initial state. We discuss the relation of Hughston’s equation to earlier work on norm-preserving stochastic equations, and show that Hughston’s equation can be written as a manifestly unitary stochastic evolution equation for the pure state density matrix. We discuss the behavior of systems constructed as direct products of independent subsystems, and briefly address the question of whether an energy-based approach, such as Hughston’s, suffices to give an objective interpretation of the measurement process in quantum mechanics.

The method of moments (MoM) is a formalization of the use of orthogonal projections in Hilbert space to obtain "best" solutions in subspaces. It is more a concept than a method. Almost any approximate solution can be interpreted according to the method of moments. This paper gives a history of the development of the method of moments as it applies to electromagnetics from the author's perspective. His learning experiences are recounted, both failures and successes. Some guesses are made as to future development and uses of the method.

The theory of angular momentum connects physical rotations and quantum spins together at a fundamental level. Physical rotation of a quantum system will therefore affect fundamental quantum operations, such as spin rotations in projective Hilbert space, but these effects are subtle and experimentally challenging to observe due to the fragility of quantum coherence. We report on a measurement of a single-electron-spin phase shift arising directly from physical rotation, without transduction through magnetic fields or ancillary spins. This phase shift is observed by measuring the phase difference between a microwave driving field and a rotating two-level electron spin system, and it can accumulate nonlinearly in time. We detect the nonlinear phase using spin-echo interferometry of a single nitrogen-vacancy qubit in a diamond rotating at 200 000rpm. Our measurements demonstrate the fundamental connections between spin, physical rotation, and quantum phase, and they will be applicable in schemes where the rotational degree of freedom of a quantum system is not fixed, such as spin-based rotation sensors and trapped nanoparticles containing spins.

0. Introduction. Many features of the theory of commutative W*-algebras in Hilbert space have been shown to extend to appropriate operator algebras in Banach spaces (cf. [2; 3; 7]). Recently an important gap in the theory was closed by Dieudonne [4] who obtained a multiplicity theory in the case that the adjoint X* of the underlying Banach space X (and hence also X) was separable. In this paper a new approach is developed which does not require assumptions of separability and allows us to construct a multiplicity theory also for the algebra of adjoints in X* where the relevant topology is the weak star or %-topology rather than the norm topology. Of the various treatments of multiplicity theory in Hilbert space our development is perhaps closest to that of Kelley [10]. The analogue of a commutative W*-algebra in a Banach space X is the operator algebra a generated in the uniform topology by a complete Boolean algebra 23 of projections in X. For given xeX, the cyclic subspace determined by x is the manifold 9Y(x) =cdm {Ax ACI }. A multiplicity function for f is a function m on e to cardinal numbers, the value m(E) being roughly the smallest number of cyclic subspaces required to span the range of E. The aim of multiplicity theory in this setting is to decompose the identity I in S into the sum of disjoint projections E of uniform multiplicity, i.e. projections such that m(F) =m(E), whenever 0 5 F<E. In ?2 such a decomposition theorem is proved to hold for any function on an abstract B.A. to cardinals which satisfies an appropriate condition of order continuity. Multiplicity functions are constructed respectively for Q3 in X anid Q3* in X* in ??3 and 6 and the cointiniuity condition is verified. Following a detailed analysis of the range of a projectioni of finite uniiform multiplicity, one of the main results of the paper (Corollary 8.5) is obtainied: A projection E in 23 has finite multiplicity n if anid oinly if its adjoint E* in Q* has multiplicity n. We have been unable to decide whether this result remains true for projections of infinite multiplicity. Finally an application is made to the theory of spectral operators of scalar tvpe dleveloped by Dunford [7; 8]. If the resolution of the identity E(.) of

Nambu quantum mechanics: a nonlinear generalization of geometric quantum mechanics

A description of differential geometrical structure of the quantum pure-states space, the projective Hilbert space, is provided. It is shown that this space is endowed with a natural symplectic structure which allows us to reformulate quantum mechanics in classical mechanical language as a classical field theory.